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Monday, May 9, 2011

Nyquist Plots (a short theory)

Nyquist Plots are a way of showing frequency responses of linear systems.  There are several ways of displaying frequency response data, including Bode' plots and Nyquist plots.
  Bode' plots use frequency as the horizontal axis and use two separate plots to display amplitude and phase of the frequency response. 
  Nyquist plots display both amplitude and phase angle on a single plot, using frequency as a parameter in the plot. 
  Nyquist plots have properties that allow you to see whether a system is stable or unstable.  It will take some mathematical development to see that, but it's the most useful property of Nyquist plots.
Nyquist Plots were invented by Nyquist - who worked at Bell Laboratories, the premiere technical organization in the U.S. at the time.  He was interested in designing telephone amplifiers to be placed in ocean-floor cables.  In those days, between the first and second world wars, undersea cables were the only reliable means of intercontinental communication.

Undersea telephone cables needed to be reliable, and to have a constant gain that did not change as the amplifier aged.  In those days, electronic amplifiers were constructed with tubes, and tubes had gains that could change dramatically as they aged.

The solution to the aging problem was to design feedback amplifiers.  However, those amplifiers could become unstable.  One morning - going to work on the Staten Island ferry, before the Verrazano Narrows bridge - Nyquist had an inspiration, and wrote his work, literally, on the back of an envelope as he rode. 

A Nyquist plot is a polar plot of the frequency response function of a linear system.

That means a Nyquist plot is a plot of the transfer function, G(s) with s = jw.  That means you want to plot G(jw).

G(jw) is a complex number for any angular frequency, w, so the plot is a plot of complex numbers.

The complex number, G(jw), depends upon frequency, so frequency will be a parameter if you plot the imaginary part of G(jw) against the real part of G(jw).

An example of a Nyquist plot will illustrate what a Nyquist plot is.
We will take a very simple system:  G(s) = 1/(s+1).
If we substitute s = jw, we get G(jw) = 1/(jw + 1).
Now, compute the real and imaginary parts of G(jw) by converting the denominator to a real number.
Now, the real part of the frequency response function is:
Real(G(jw)) = 1/(1+w2)
And, the imaginary part is:
Imag(G(jw)) = jw/(1+w2)
- or you may prefer that we express this as:
 Imag(G(jw)) = w/(1+w2) - leaving off the j.
        Now, to generate a Nyquist plot we would need to plot the imaginary part on the vertical axis of a plot, and the real part on the horizontal axis.  Here is a video of that operation.
        The point at which the phase angle becomes -45o is important.  You can read the frequency from the clip.  Determine the frequency on the clip at which the phase is closest to -45o.
        Now, since the transfer function, G(s), is 1/(s + 1) for this example, we can determine what should have been the answer, not just the closest frame on the video.  Let's determine the frequency at which the phase angle is -45o.
  • The frequency response function is G(jw) = 1/(jw + 1).
  • The phase angle is -45o when the angle of the denominator is +45o.
  • The angle of the denominator is tan-1(w).
  • Solving for the frequency, w, we get w = 1.0.
  • If w = 1.0, then f =  w/2p = .159 Hz.
        The video of the Nyquist plot isn't really a true Nyquist plot.  A true Nyquist plot shows the frequency response function for all frequencies, not just a single -albeit moving - point.  So, let's take a look at the Nyquist plot for G(s) = 1/(s + 1).  Here it is!
Now, let us look at some interesting points in this Nyquist plot.
  • The low frequency portion of the plot is near +1.  That makes sense since the DC gain is 1 for G(s) = 1/(s + 1).
  • The high frequency portion of the plot is near the origin in the G(jw) plane.  That makes sense because the magnitude becomes small as frequency gets large.
  • The high frequency portion of  the plot approaches the origin at an angle of -90o.  That makes sense because the phase approaches -90o as the frequency gets large.
        What's wrong with all of this?  Is there something else we should note?
  • The frequency is a parameter of the plot, and unless we do something, there will be no indication of what frequency corresponds to a particular point on the plot.
  • We can indicate direction of increase of frequency with small arrows along the plot.  Using those arrows is  more- or-less standard practice.
  • We will always assume that the plot starts at zero frequency and frequency goes to infinity.
        At this point, you have seen one Nyquist plot.  We need to consider a few more points about Nyquist plots.
  • You need to learn what Nyquist plots look like for different systems, including second order systems, higher order systems, systems with resonant peaks and systems with poles and zeroes at the origin of the s-plane.
  • You need to learn how you can generate Nyquist plots.
High Frequency Asymptotes        There are other points you need to note about Nyquist plots. Let's start by considering how a Nyquist plot is affected when the system has a higher order.

  • First, consider a more general transfer function.  Most transfer functions are a ratio of polynomials in s.  Here is a typical example - shown in factored form.
    • This system has m zeroes.
    • This system has n poles.
  • It is almost always true that the denominator is of higher order than the numerator so,
n  > m, i.e.#Poles > # Zeroes
  • Although, on occasion we have:
n = m, i.e.#Poles = # Zeroes
  • The system has n poles and m zeroes.
  • We remind you that a stable system will have all of the poles in the left half of the s-plane, so all of the p's will be negative in G(s).  Also, zeroes will usually be in the left half of the s-plane, but it's possible that is not the case.
        Now, let us assume - at least for the moment - that:
  • The transfer function has no poles at s = 0.  We normally say that the system has no poles at the origin.
  • We are going to examine the behavior of the Nyquist plot for large frequencies.
  • Let s = jw in the transfer function to obtain:
Then, if we let the frequency become very large.  In the limit, each jw term will "overpower" the corresponding z or p term in G(jw) and we will have:
G(jw) ~= 1/(jwn-m)
        The angle of this limiting form is what we are interested in now, and the angle is determined by the j-term.
  • The angle is determined by the power of j.  You get -90o for every j.
  • For example, if n = 4, and m = 1, then n - m = 3, and for high frequencies the Nyquist plot would have an angle of -270o.
        Here are some examples.  For each example, think about the asymptotes, then click on the hot word or the Nyquist plot to show the high frequency asymptote when you have determined what the angle should be.
  • A first order system, G(s) = 1/(s + 1)
    • Click on the button to see the high frequency asymptote.
    • The high frequency asymptote is at -90o which is where it should be for a system with one more pole than zero.
  • A second order system, G(s) = 1/(s + 1)2
    • Click on the button to see the high frequency asymptote.
    • The high frequency asymptote is at -180o which is where it should be for a system with two more poles than zeroes.
  • A third order system, G(s) = 1/(s + 1)3
    • Click on the button to see the high frequency asymptote.
    • The high frequency asymptote is at -270o which is where it should be for a system with three more poles than zeroes.
        The example third order system is not easily seen.  However, you can change the scale for that system, and see things more clearly.  If you have a problem seeing the asymptote you may want to change scales when you have to do this kind of analysis.

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